Antiderivatives

1. Introduction
Calculus is the study of change.
In both of these branches (Differential and Integral), the concepts learned in algebra
and geometry are extended using the idea of limits. Limits allow us to study what happens
when points on a graph get closer and closer together until their distance is infinitesimally
small (almost zero). Once the idea of limits is applied to our Calculus problem, the
techniques used in algebra and geometry can be implemented.
Differential calculus and integral calculus are connected by the fundamental
theorem of calculus, which states that differentiation is the reverse process to integration.
Leibniz’s Creation of the Calculus
In the years from 1672 to 1676, spent in Paris, Leibniz’s slowly flowering
mathematical genius matured. During this time, he developed the principal features and
notation of his version of the Calculus. Various methods had been invented for determining
the tangent lines to certain classes of curves, but as yet nobody had made known similar
procedures for solving the inverse problem, that is deriving the equation of the curve itself
from the properties of the tangents. Leibniz stated the inverse tangent problem thus: “To
find the locus of the function, provided the locus which determines the subtangent is
known.” By the middle of 1673, he had settled down to an exploration of this problem, fully
recognizing that “almost the whole of the theory of the inverse method of tangents is
reducible to quadratures [integrations].”

2. Because Leibniz was still struggling with the notation for his calculus, it is not
surprising that these early calculations were clumsy. Either he expressed his results in
rhetorical form or else used abbreviations, such as “omn.” For the Latin omnia (“all”) to
mean “sum.” The letter l was used to symbolize what we should write as dy, the
“difference” of two neighboring ordinates .
Differential Calculus
In mathematics, differential calculus is a subfield of calculus concerned with the
study of the rates at which quantities change. It is one of the two traditional divisions of
calculus, the other being integral calculus.
The primary objects of study in differential calculus are the derivative of a function,
related notions such as the differential, and their applications. The derivative of a function
at a chosen input value describes the rate of change of the function near that input value.
The process of finding a derivative is called differentiation. Geometrically, the derivative at a
point is the slope of the tangent line to the graph of the function at that point, provided that
the derivative exists and is defined at that point. For a real-valued function of a single real
variable, the derivative of a function at a point generally determines the best linear
approximation to the function at that point.
Integral Calculus
The integral is an important concept in mathematics. Integration is one of the two
main operations in calculus, with its inverse, differentiation, being the other. Given a
function f of a real variable x and an interval [a, b] of the real line, the definite integral
is defined informally as the signed area of the region in the xy-plane that is bounded
by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis
adds to the total and that below the x-axis subtracts from the total.
The term integral may also refer to the related notion of the antiderivative, a
function F whose derivative is the given function f. In this case, it is called an indefinite
integral and is written:

3. What is the difference between these two?
 Differential Calculus cuts something into small pieces to find how it changes.
 Integral Calculus joins (integrates) the small pieces together to find how much there
is.
Antidifferentiation
Antidifferentiation is the process of finding the set of all the antiderivatives of a
given function. The symbol ∫ denotes the operation of antidifferentiation, and we write
∫ 푓(푥)푑푥 = 퐹(푥) + 퐶
where 퐹′ (푥) = 푓(푥) and 푑(퐹(푥)) = 퐹(푥) + 퐶. The expression 퐹(푥) + 퐶 is the general
antiderivative of f.
Leibniz introduced the convention of writing the differential of a function after the
antidifferentiation symbol. The advantage of using the differential in this manner will be
apparent when we compute antiderivatives by changing the variable. From the above
equations, we can write:
∫ 푑(퐹(푥)) = 퐹(푥) + 퐶
This equation states that when we antidiffentiate the differential of a function, we obtain
that function plus an arbitrary constant. So we can think of the ∫ symbol for
antidifferentiation as meaning that operation which is the inverse of the operation denoted
by d for computing differential.
If { 퐹(푥) + 퐶} is the set of all functions whose differentials are푓(푥)푑푥, it is also the
set of all functions whose derivatives are f(x). Antidifferentiation, therefore, is considered as
the operation of finding the set of all functions having a given derivative.
Notation for Antiderivatives
The antidifferentiation process is also called integration and is denoted by the
symbol ∫ called an integral sign. The symbol ∫ 푓(푥)푑푥 is called the indefinite integral of f(x),
and it denotes the family of antiderivatives of f(x). That is, if 퐹′ (푥) = 푓(푥) for all x, then
∫ 푓(푥)푑푥 = 퐹(푥) + 퐶 where f(x) is called the integrand and C the constant of integration.
The differential dx in the indefinite integral identifies the variable of integration. That
is, the symbol ∫ 푓(푥)푑푥 denotes the “antiderivative of f with respect to x” just as the symbol
dy/dx denotes the “derivative of y with respect to x.”

4. Integration Tables
Basic Integration Formulas
Rules Antiderivatives
1. Constant Rule ∫ 푑푥 = 푥 + 퐶
2. Simple Power Rule
∫ 푥 푛 푑푥 =
푥 푛+1
푛 + 1
+ 퐶, 푛 ≠ −1
3. General Power Rule
∫ 푢푛
푑푢
푑푥
푑푥 = ∫ 푢푛 푑푢 =
푢푛+1
푛 + 1
+ 퐶, 푛 ≠ −1
4. Simple Log Rule
∫
1
푥
푑푥 = ln|푥| + 퐶
5. General Log Rule
∫
1
푢
푑푢
푑푥
푑푥 = ∫
1
푢
푑푢 = ln|푢| + 퐶
6. Simple Exponential Rule ∫ 푒푥 푑푥 = 푒푥 + 퐶
7. General Exponential Rule
∫ 푒푢 푑푢
푑푥
푑푥 = ∫ 푒푢 푑푢 = 푒푢 + 퐶
Trigonometric Functions Integration Formulas
Functions Antiderivatives
1. ∫ 푐표푠푥 푑푥 푠푖푛푥 + 퐶
2. ∫ 푠푖푛푥 푑푥 −푐표푠푥 + 퐶
3. ∫ 푠푒푐2 푥 푑푥 푡푎푛푥 + 퐶
4. ∫ 푐푠푐2 푥 푑푥 −푐표푡푥 + 퐶
5. ∫ 푠푒푐푥푡푎푛푥 푑푥 푠푒푐푥 + 퐶
6. ∫ 푐푠푐푥푐표푡푥 푑푥 −푐푠푐푥 + 퐶

5. Sample Problems Solving the Antiderivatives of Basic Equation

6. Sample Problems Solving the Antiderivatives of Trigonometric Functions

7. Differential Equations
A differential equation is an equation involving a function and one or more of its
derivatives. For instance,
3
푑푦
푑푥
− 2푥푦 = 0
is a differential equation in which y = f(x) is a differentiable function of x.
A function y = f(x) is called a solution of a given differential equation if the equation is
satisfied when y and its derivatives are replaced by f(x) and its derivatives. For example,
푦 = 푒−2푥
is a solution to the differential equation
푦′ + 2푦 = 0
because 푦′ = −2푒−2푥 , and by substitution we have
−2푒−2푥 + 2푒−2푥 = 0
Furthermore, we can readily see that
푦 = 2푒−2푥 , 푦 = 3푒−2푥 , 푦 = 1
2
푒−2푥
are also solutions to the same differential equation. In fact, the functions
푦 = 퐶푒−2푥
where C is any real number, are all solutions. We call this family of solutions the general
solution of the differential equation 푦′ + 2푦 = 0.
A particular solution of a differential equation is any solution that is obtained by
assigning specific values to the constants in the general solution.
Geometrically, the general solution of a given differential equation represents a
family of curves, called solution curves one for each value assigned to the arbitrary
constants.

8. Example of Differential Equations (First and Second Order)
1. Show that
푑2 푦
푑푥2 = 2 푑푦
푑푥
a solution of y = c1 + c2e2x
Solution:
Since y = c1 + c2e2x, then:
푑푦
푑푥
= 2c2e2x
and
2 푑푦
푑푥
= 4c2e2x
Therefore,
푑2 푦
푑푥2 = 2 푑푦
푑푥
.
References
Books
 Burton, David M. The History of Mathematics
 Larson and Hostetler. Brief Calculus with Applications. “Integration.” Fourth Printing.
P 316-518
 Leithold, Louis. TC 7 (The Calculus 7). P 315-32
Websites
http://en.wikipedia.org/wiki/Calculus
http://en.wikipedia.org/wiki/Integral
http://en.wikipedia.org/wiki/Differential_calculus
http://www.mathsisfun.com/calculus/integration-introduction.html
http://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx